Question

a price of a bow Brand A crackers was x percent more than the price of a box of Brand B sydney paid 455.20 for some boxes of Brand A crackers. If she bought the same number of boxes of Brand B crackers instead, she would pay $7.20 less. What was the value of x.

Answer 1

the price of a box of Brand B crackers is P, then the price of a box of Brand A crackers would be (100+X)/100 * P. According to the problem, Sydney paid 455.20 for some boxes of Brand A crackers. If she had bought the same number of boxes of Brand B crackers instead, she would have paid $7.20 less. This means: (number of boxes) * [(100+X)/100 * P] = (number of boxes) * P - 7.20 We can simplify and solve for X: (number of boxes) * P * X/100 = 7.20 X = 720 / (number of boxes * P) Since we don't have enough information about the value of P, we can't find the exact value of X.

Answer 2

Let's start by setting up an equation for the price of the boxes of Brand A crackers that Sydney bought:

Price of Brand A crackers = Price of Brand B crackers * (1 + x/100)

We don't know the value of x, but we do know that if Sydney had bought the same number of boxes of Brand B crackers instead, she would have paid $7.20 less. This means that:

Price of Brand B crackers - $7.20 = Price of Brand A crackers

We can substitute the first equation into the second equation to eliminate the price of Brand A crackers:

Price of Brand B crackers - $7.20 = Price of Brand B crackers * (1 + x/100)

Simplifying this equation, we get:

$7.20 = Price of Brand B crackers * (x/100)

Now we can solve for x:

x = ($7.20 / Price of Brand B crackers) * 100

We don't know the price of Brand B crackers, but we do know that Sydney paid $455.20 for some boxes of Brand A crackers. Let's assume she bought n boxes of Brand A crackers, so the price of each box is:

Price of Brand A crackers = $455.20 / n

We can substitute this into the equation for x to get:

x = ($7.20 / ($455.20 / n)) * 100

Simplifying this equation, we get:

x = (n /$63.00) - 100

Now we need to find the value of n. We know that if Sydney had bought the same number of boxes of Brand B crackers instead, she would have paid $7.20 less. So:

Price of Brand A crackers - $7.20 = Price of Brand B crackers

$455.20 / n - $7.20 = Price of Brand B crackers

We can substitute this into the equation for x to get:

x = (n / $63.00) - 100

x = ((455.20 / (Price of Brand B crackers + $7.20)) / $63.00) - 100

Multiplying both sides by $63.00, we get:

x * $63.00 = ((455.20 / (Price of Brand B crackers + $7.20))) - $6300

Simplifying the right-hand side, we get:

x * $63.00 = (455.20 / (Price of Brand B crackers + $7.20)) - $6300

Multiplying both sides by (Price of Brand B crackers + $7.20), we get:

x * $63.00 * (Price of Brand B crackers + $7.20) = 455.20 - $63.00 * (Price of Brand B crackers + $7.20)

Expanding and simplifying, we get:

x * $63.00 * Priceof Brand B crackers + x * $63.00 * $7.20 = $455.20 - $63.00 * Price of Brand B crackers - $63.00 * $7.20

x * $63.00 * Price of Brand B crackers + $455.40 = $455.20 - $63.00 * Price of Brand B crackers - $453.60

x * $126.00 * Price of Brand B crackers = $1.60

x = $1.60 / ($126.00 * Price of Brand B crackers)

We don't know the price of Brand B crackers, so we can't solve for x yet. However, we can simplify the expression for x:

x = 0.0126984 / Price of Brand B crackers

Now we need to find the price of Brand B crackers. We know that Sydney paid $455.20 for some boxes of Brand A crackers, so let's assume she bought n boxes of Brand A crackers, so the price of each box is:

Price of Brand A crackers = $455.20 / n

We also know that if she had bought the same number of boxes of Brand B crackers instead, she would have paid $7.20 less. So:

Price of Brand A crackers - $7.20 = Price of Brand B crackers

Substituting the expression for Price of Brand A crackers, we get:

$455.20 / n - $7.20= Price of Brand B crackers

We can substitute this into the expression for x to get:

x = 0.0126984 / ($455.20 / n - $7.20)

Simplifying the denominator, we get:

x = 0.0126984 / (($455.20 - $7.20n) / n)

Multiplying both sides by n, we get:

x * n = 0.0126984 / (($455.20 - $7.20n) / n) * n

Simplifying, we get:

x * n = 0.0126984 / ($455.20 - $7.20n) * n^2

Multiplying both sides by ($455.20 - $7.20n), we get:

x * n * ($455.20 - $7.20n) = 0.0126984 * n^2

Expanding and simplifying, we get:

-0.00091n^3 + $3.6448n^2 - $312.624n + $7.66976 = 0

We can solve this cubic equation for n using numerical methods like the cubic formula or Newton's method. However, the exact value of n is not needed to find x. We can use a graphical method or a numerical method like the bisection method to find an approximate value of n that satisfies the equation. Once we have an approximate valueof n, we can substitute it into the equation for Price of Brand A crackers and the equation for Price of Brand B crackers, and then into the expression for x to find the value of x.

For example, let's use the bisection method to find an approximate value of n. We know that n must be greater than 0 and less than or equal to $63.20 / ($7.20 + Price of Brand B crackers), because Sydney paid $7.20 less for the same number of boxes of Brand B crackers. Let's start with the midpoint of this interval, which is:

n = $63.20 / ($7.20 + $31.60) = 1.6

We can substitute this into the equation for Price of Brand A crackers and the equation for Price of Brand B crackers to get:

Price of Brand A crackers = $455.20 / 1.6 = $284.50

Price of Brand B crackers = $284.50 - $7.20 = $277.30

We can substitute these into the expression for x to get:

x = ($7.20 / $277.30) * 100 = 2.6 (rounded to one decimal place)

Therefore, the value of x is approximately 2.6%.

Answer 3

Sydney paid $455.20 for some boxes of Brand A crackers, which were x percent more expensive than the boxes of Brand B crackers. If she had bought the same number of boxes of Brand B crackers instead, she would have paid $7.20 less. Now we need to find out the value of x.